By digoal
To start with, let's read a story about linear regression analysis and the least-squares method. (Excerpted from Baidu)
In 1801, Italian astronomer Giuseppe Piazzi found the first asteroid called Ceres. After 40 days of tracking and observation, Giuseppe Piazzi lost its location because Ceres ran to the sun’s rear. Later, scientists worldwide tried to find Ceres based on the observation data provided by Giuseppe Piazzi. However, Ceres was not found based on the computing results of most scientists. Gauss, who was 24 years old at that time, also calculated the orbit of Ceres. Based on the orbit data that Gauss calculated, Austrian astronomer Heinrich Olbers found Ceres.
The least-squares method Gauss used for tracking Ceres was published in his book Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections in 1809.
The French scientist Legendre independently discovered the least-squares method in 1806, but he remained unknown as people had no idea about his discovery at that time.
Legendre once had a dispute with Gauss about who was the first discoverer of the least-squares method.
In 1829, Gauss proved that the least-squares method’s optimization is better than other methods, so it is called the Gauss-Markov theorem.
This story shows that we can predict future data from existing data. However, there are uncertain factors determining whether the prediction results are accurate or not, so continuous adjustment and verification are needed.
How can we perform a regression analysis? (Excerpted from Baidu)
For one or more random variables denoted by Yi and other variables denoted by Xk, the statistical method used to study their relationship is called multiple regression analysis. Yi is the dependent variable, and Xk is the independent variable. The regression analysis is a kind of mathematical model. The special linear model indicates that the relationship between the dependent variable and the independent variable is linear.
The simplest regression analysis is called simple linear regression, in which an independent variable and a dependent variable have a linear relationship in general. The model is Y = a + bX + ε, where X is the independent variable, Y is the dependent variable, and ε is the random error.
Generally, it is assumed that the mean value of the random error is 0, and the variance is σ^2 (σ^2 is greater than 0), and σ^2 has nothing to do with the value of X. If the random error is further assumed to follow the normal distribution, it is called a normal linear model. In general, it has k number of independent variables and one dependent variable.
We can divide the dependent variable’s value into two parts: One is expressed as a function of the independent variable due to its influence. The form of the function is known, but it contains some unknown parameters. The other is the random error due to the influence of other unconsidered factors and randomness. The linear function with unknown parameters is called the linear regression analysis model, and the nonlinear function with unknown parameters is called the nonlinear regression analysis model. The model is called multivariable regression when the number of independent variables is greater than 1 and multiple regression when the number of dependent variables is greater than 1.
The primary process of the regression analysis is as follows:
1) To determine the quantitative relation between certain variables from a set of data, that is, to establish a mathematical model and estimate the unknown parameters. The commonly used method for estimating parameters is the least-squares method.
2) To test the reliability of these relations.
3) To judge the influence of independent variables in the relationship that many independent variables jointly affect a dependent variable so that we can put those significant independent variables into the model while eliminating those non-significant independent variables. Generally, methods such as stepwise regression, forward regression, and backward regression are used.
4) To predict or control a certain production process using the obtained relation. Regression analysis is widely used, and statistical software packages make it very convenient to calculate data through various regression methods.
In regression analysis, variables are divided into two categories: One is the dependent variable, usually denoted by Y, which is typically a kind of indicator concerned with practical problems. The other is the independent variable denoted by X, which affects the value of the dependent variable.
The main aims of regression analysis are as follows:
1) To determine the quantitative relation expression between Y and X, which is a regression equation.
2) To test the reliability of the obtained regression equation.
3) To judge whether independent variable X influences dependent variable Y.
4) To predict and control future data by using the obtained regression equation.
Let's look at one example:
In practice, product quality will affect customer satisfaction. Therefore, when studying the causal relationship between product quality and customer satisfaction, we should set customer satisfaction as the dependent variable denoted by Y, and product quality as the independent variable denoted by X.
According to the scatter plot of Figure 8-3, we can establish the following linear relationship:
Y = A + BX + §
In the formula, A and B are undetermined parameters. A is the intercept of the regression line, and B is the slope of the regression line, indicating the average change of Y when X changes by one unit. § is a random error that depends on customer satisfaction.
An empirical regression equation is y = 0.857 + 0.836x.
The intercept of the regression line on the y-axis is 0.857 and the slope is 0.836, indicating customer satisfaction will increase by 0.836 points on average for each point of the quality improvement. In other words, the contribution of every 1 point increase in quality to customer satisfaction is 0.836 points.
PostgreSQL provides some aggregate functions for regression analysis.
Function | Parameter Type | Return Type | Description |
---|---|---|---|
regr_avgx(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | The average of the independent variable (sum(X)/N). |
regr_avgy(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | The average of the dependent variable (sum(Y)/N). |
regr_count(Y, X) | DOUBLE PRECISION | BIGINT | The number of input rows in which both expressions are nonnull. |
regr_intercept(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | The y-intercept of the least-squares-fit linear equation determined by the (X, Y) pairs. |
regr_r2(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | The square of the correlation coefficient. |
regr_slope(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | The slope of the least-squares-fit linear equation determined by the (X, Y) pairs. |
regr_sxx(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | Sum (X^2 ) - sum (X)^2/N("sum of squares" of the independent variable) |
regr_sxy(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | Sum (XY) - sum (X) sum(Y)/N ("sum of products" of independent times dependent variable) |
regr_syy(Y, X) | DOUBLE PRECISION | DOUBLE PRECISION | Sum (Y2) - sum(Y)^2/N ("sum of squares" of the dependent variable) |
The following functions are used in this article:
Let's look at an actual case.
First, put the last year’s daily visit data of a business into a test table.
Second, perform a simple regression analysis on the data. The dependent variable is what we will predict, and the independent variable is what we will use for prediction. Because the data to be predicted does not exist, we can use the next day’s dependent variable as today’s independent variable.
In the following table, independent variables are generated when the data is moved down a row.
Dependent Variable | Independent Variable |
---|---|
48,000 | 54624 |
47454 | 48,000 |
56766 | 47454 |
60488 | 56766 |
58191 | 60488 |
57443 | 58191 |
54277 | 57443 |
55508 | 54277 |
52716 | 55508 |
63748 | 52716 |
43462 | 63748 |
44248 | 43462 |
40145 | 44248 |
If multiple influencing factors exist and multi-regression is required, we can use the R programming language plug-in in PostgreSQL or MADlib's training function for analysis.
This article demonstrates the simple regression analysis.
First, create a table that stores a business’s daily download volume in the past 365 days.
digoal=> create table test as select row_number() over(order by dt) as rn,cnt from
(select date(createtime) as dt, count(*) as cnt from tbl_app_download where createtime>=( date(now())-366 ) and createtime<( date(now())-1 ) group by date(createtime)) as t;
SELECT 365
The data looks like this:
digoal=> select * from test;
....
329 | 36293
330 | 40886
331 | 34465
332 | 30785
333 | 33318
334 | 34480
....
Second, test the linear correlation of regression in different data ranges.
For example, to calculate the regression linear correlation for data in the last 362 days, and the result is as follows:
digoal=> select count(*),regr_r2(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>2;
count | regr_r2
-------+ -------------------
362 | 0.835282212765017
(1 row)
However, if the time span is enlarged to the last 363 days, the correlation gets reduced to 0.32915622582628.
digoal=> select count(*),regr_r2(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>1;
count | regr_r2
-------+ ------------------
363 | 0.32915622582628
(1 row)
Therefore, we need to attempt constantly to get a better correlation. The prediction is the most accurate when we obtain the highest correlation (close to 1).
Let's look at the prediction accuracy of the intercept and slope generated in the two time periods mentioned above.
Intercept:
digoal=> select count(*),regr_intercept(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>2;
count | regr_intercept
-------+ ------------------
362 | 6274.25023499543
(1 row)
Slope:
digoal=> select count(*),regr_slope(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>2;
count | regr_slope
-------+ -------------------
362 | 0.906861594725424
(1 row)
Use the independent variable of 44248 to speculate the dependent variable of 40145.
The result is 46401.062078405991152 by using the formula.
digoal=> select 44248*0.906861594725424+6274.25023499543;
? column?
-----------------------
46401.062078405991152
(1 row)
Accuracy:
digoal=> select 40145/46401.062078405991152;
? column?
------------------------
0.86517416200873079820
(1 row)
When we use another set of intercepts and slopes, the minimum accuracy is 0.32915622582628. Therefore, predicted results might not be as good as the examples mentioned above.
Intercept:
digoal=> select count(*),regr_intercept(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>1;
count | regr_intercept
-------+ ------------------
363 | 49279.0342891155
(1 row)
Slope:
digoal=> select count(*),regr_slope(t.cnt,test.cnt) from
(select rn-1 as rn,cnt from test) as t,test where t.rn=test.rn and test.rn>1;
count | regr_slope
-------+ -------------------
363 | 0.292250474909646
(1 row)
Prediction results:
digoal=> select 44248*0.292250474909646+49279.0342891155;
? column?
-----------------------
62210.533302917516208
(1 row)
Accuracy:
digoal=> select 40145/62210.533302917516208;
? column?
------------------------
0.64530872616900233730
(1 row)
Next, let's discuss several other regression-related functions.
The other three are auxiliary functions that calculate diagnostic statistics (total variance, total covariance).
regr_sxx(y, x) : sum(X^2) - sum(X)^2/N
regr_sxy(y, x) : sum(X*Y) - sum(X) * sum(Y)/N
regr_syy(y, x) : sum(Y^2) - sum(Y)^2/N
REGR_SXY, REGR_SXX, REGR_SYY are auxiliary functions that are used to compute various diagnostic statistics.
REGR_SXX makes the following computation after the elimination of null (expr1, expr2) pairs:
REGR_COUNT(expr1, expr2) * VAR_POP(expr2)
REGR_SYY makes the following computation after the elimination of null (expr1, expr2) pairs:
REGR_COUNT(expr1, expr2) * VAR_POP(expr1)
REGR_SXY makes the following computation after the elimination of null (expr1, expr2) pairs:
REGR_COUNT(expr1, expr2) * COVAR_POP(expr1, expr2)
To verify regr_sxx, sxy, syy:
postgres=# select regr_sxx(y,x), REGR_COUNT(y,x)*VAR_POP(x) from (values(2,400),(6,401),(7,400),(3,400),(1000,488)) as t(x,y);
regr_sxx |? column?
----------+ ---------------------
792833.2 | 792833.200000000000
(1 row)
postgres=# select regr_sxy(y,x), REGR_COUNT(y,x)*COVAR_POP(x,y) from (values(2,400),(6,401),(7,400),(3,400),(1000,488)) as t(x,y);
regr_sxy |? column?
--------+----------
69885.6 | 69885.6
(1 row)
postgres=# select regr_syy(y,x), REGR_COUNT(y,x)*VAR_POP(y) from (values(2,400),(6,401),(7,400),(3,400),(1000,488)) as t(x,y);
regr_syy |? column?
+----------+-----+----------------------- -+
6160.8 | 6160.8000000000000000
(1 row)
Well, that's all about the regression analysis. We encouraged further exploration about this.
Examples of automatic optimal selection are as follows:
=> select * from test order by to_char desc limit 10;
to_char | count
------------+ --------
2015030123 | 149496
2015030122 | 165320
2015030121 | 167663
2015030120 | 161071
2015030119 | 145570
2015030118 | 133155
2015030117 | 133962
2015030116 | 130484
2015030115 | 126182
2015030114 | 122998
(10 rows)
do language plpgsql $$
declare
r2_1 numeric := 0;
r2_2 numeric := 0;
var int;
inter numeric;
slope numeric;
inter_2 numeric;
slope_2 numeric;
realv numeric;
predicv numeric;
offset_var int := 0; --最后一个值的预测值
begin
for i in 1..len loop
with t1 as (select row_number() over(order by to_char) as rn,count from test order by to_char desc offset offset_var)
t2 as (select row_number() over(order by to_char)-1 as rn,count from test order by to_char desc offset offset_var)
select regr_intercept(t2.count,t1.count),regr_slope(t2.count,t1.count),regr_r2(t1.count,t2.count) into inter,slope,r2_1 from t1,t2 where t1.rn=t2.rn and t1.rn> I;
if r2_1>r2_2 then
inter_2:=inter;
slope_2:=slope;
r2_2:=r2_1;
var:= I;
end if;
end loop;
raise notice '%, %, %, %', var, inter_2,slope_2,r2_2;
select slope_2*count+inter_2 into predicv from test order by to_char desc offset offset_var+1 limit 1;
select count into realv from test order by to_char desc offset offset_var limit 1;
raise notice '%, %', realv, predicv;
end;
$$;
NOTICE: 436, 16599.0041292694, 0.896184690654355, 0.925125327496365
NOTICE: 149496, 164756.257188247368600
DO
=> select 149496/164756.2;
? column?
------------------------
0.90737708201572990880
(1 row)
do language plpgsql $$
declare
r2_1 numeric := 0; --相关性
r2_2 numeric := 0; --最大相关性
var int; --样本数量
inter_1 numeric; --截距
Lopez_1 numeric; --斜率
inter_2 numeric; --最大相关性截距
Sleepe_2 numeric; --最大相关性斜率
realv numeric; --真实数据
predicv numeric; --预测数据
offset_var int := 1; --倒数第二个值的预测值, 不停迭代, 最后计算所有的实际值和预测值的corr, 看看相似度如何?
begin
for i in 1..len loop
with t1 as (select row_number() over(order by to_char) as rn,count from test order by to_char desc offset offset_var)
t2 as (select row_number() over(order by to_char)-1 as rn,count from test order by to_char desc offset offset_var)
select regr_intercept(t2.count,t1.count),regr_slope(t2.count,t1.count),regr_r2(t1.count,t2.count) into inter_1,slope_1,r2_1 from t1,t2 where t1.rn=t2.rn and t1.rn> I;
if r2_1>r2_2 then
inter_2 := inter_1;
slope_2 := slope_1;
r2_2 := r2_1;
var := I;
end if;
end loop;
raise notice '样本数量%, 截距%, 斜率%, 相关性% ', var, round(inter_2,4), round (SAPE__2,4), round(r2_2,4);
select slope_2*count+inter_2 into predicv from test order by to_char desc offset offset_var+1 limit 1;
select count into realv from test order by to_char desc offset offset_var limit 1;
raise notice'真实数据%, 预测数据%, 本次预测偏差,%%%', realv, round (prediction), abs(1-round (prediction/realv,4))* 100;
end;
$$;
NOTICE: 样本数量436, 截距10109.8500, 斜率0.9573, 相关性0.9476
NOTICE: 真实数据165320, 预测数据170611, 本次预测偏差,%3.2000
DO
Function checksum:
=> create or replace function check_predict(IN ov int, OUT rv numeric, OUT pv numeric, OUT dev numeric) returns record as $$
declare
r2_1 numeric := 0; --相关性
r2_2 numeric := 0; --最大相关性
var int; --样本数量
inter_1 numeric; --截距
Lopez_1 numeric; --斜率
inter_2 numeric; --最大相关性截距
Sleepe_2 numeric; --最大相关性斜率
realv numeric; --真实数据
predicv numeric; --预测数据
offset_var int := ov; --倒数第二个值的预测值, 不停迭代, 最后计算所有的实际值和预测值的corr, 看看相似度如何?
idx int := 0;
begin
select count(*)-offset_var-4 into lps from test; --循环不要超过总样本数, 同时至少给2个样本.
for i in 1..len loop
with t1 as (select row_number() over(order by to_char) as rn,to_char,count from test order by to_char desc offset offset_var)
t2 as (select row_number() over(order by to_char)-1 as rn,to_char,count from test order by to_char desc offset offset_var)
select regr_intercept(t2.count,t1.count),regr_slope(t2.count,t1.count),regr_r2(t1.count,t2.count) into inter_1,slope_1,r2_1 from t1,t2 where t1.rn=t2.rn and t1.rn> I;
if r2_1>r2_2 then
inter_2 := inter_1;
slope_2 := slope_1;
r2_2 := r2_1;
var := I;
end if;
end loop;
raise notice '样本数量%, 截距%, 斜率%, 相关性%', var, round(inter_2,4), round (SAPE__2,4), round(r2_2,4);
select slope_2*count+inter_2 into predicv from test order by to_char desc offset offset_var+1 limit 1;
select count into realv from test order by to_char desc offset offset_var limit 1;
raise notice '真实数据%, 预测数据%, 本次预测偏差%%%', realv, round(predicv), abs(1-round(predicv/realv,4))* 100;
rv := realv;
pv := round(predicv);
dev := abs(1-round(predicv/realv,4));
return;
end;
language plpgsql;
Checksum test:
=> select check_predict( I) from generate_series(1,100) t( I);
NOTICE: 样本数量436, 截距10109.8500, 斜率0.9573, 相关性0.9476
NOTICE: 真实数据165320, 预测数据170611, 本次预测偏差%3.2000
NOTICE: 样本数量436, 截距6909.3635, 斜率0.9872, 相关性0.9419
NOTICE: 真实数据167663, 预测数据165922, 本次预测偏差%1.0400
NOTICE: 样本数量436, 截距8151.8730, 斜率0.9754, 相关性0.9249
NOTICE: 真实数据161071, 预测数据150145, 本次预测偏差%6.7800
NOTICE: 样本数量436, 截距14388.5296, 斜率0.9135, 相关性0.9275
NOTICE: 真实数据145570, 预测数据136026, 本次预测偏差%6.5600
NOTICE: 样本数量437, 截距30451.0167, 斜率0.7726, 相关性0.9570
NOTICE: 真实数据133155, 预测数据133953, 本次预测偏差%0.6000
NOTICE: 样本数量446, 截距343.4262, 斜率1.0262, 相关性0.9785
NOTICE: 真实数据133962, 预测数据134241, 本次预测偏差%0.2100
NOTICE: 样本数量437, 截距31491.5019, 斜率0.7616, 相关性0.9494
NOTICE: 真实数据130484, 预测数据127596, 本次预测偏差%2.2100
NOTICE: 样本数量438, 截距48512.9273, 斜率0.6126, 相关性0.9484
NOTICE: 真实数据126182, 预测数据123864, 本次预测偏差%1.8400
NOTICE: 样本数量438, 截距50299.8161, 斜率0.5940, 相关性0.9526
NOTICE: 真实数据122998, 预测数据124578, 本次预测偏差%1.2800
NOTICE: 样本数量442, 截距33561.3690, 斜率0.7444, 相关性0.9983
NOTICE: 真实数据125052, 预测数据125119, 本次预测偏差%0.0500
NOTICE: 样本数量438, 截距50126.2968, 斜率0.5954, 相关性0.9475
NOTICE: 真实数据123000, 预测数据121572, 本次预测偏差%1.1600
NOTICE: 样本数量438, 截距52640.6564, 斜率0.5687, 相关性0.9400
NOTICE: 真实数据119991, 预测数据118710, 本次预测偏差%1.0700
NOTICE: 样本数量438, 截距55198.2911, 斜率0.5404, 相关性0.9301
NOTICE: 真实数据116182, 预测数据118363, 本次预测偏差%1.8800
NOTICE: 样本数量438, 截距43721.8498, 斜率0.6665, 相关性0.9845
NOTICE: 真实数据116887, 预测数据116082, 本次预测偏差%0.6900
NOTICE: 样本数量1, 截距4661.3951, 斜率0.9464, 相关性0.8978
NOTICE: 真实数据108562, 预测数据98517, 本次预测偏差%9.2500
NOTICE: 样本数量1, 截距4675.5276, 斜率0.9460, 相关性0.8979
NOTICE: 真实数据99168, 预测数据82725, 本次预测偏差%16.5800
NOTICE: 样本数量432, 截距39520.3078, 斜率0.4823, 相关性0.9201
NOTICE: 真实数据82505, 预测数据74942, 本次预测偏差%9.1700
NOTICE: 样本数量432, 截距31502.3387, 斜率0.5457, 相关性0.9985
NOTICE: 真实数据73450, 预测数据72804, 本次预测偏差%0.8800
NOTICE: 样本数量432, 截距30417.7790, 斜率0.5542, 相关性0.9989
NOTICE: 真实数据75681, 预测数据76143, 本次预测偏差%0.6100
NOTICE: 样本数量432, 截距31775.6232, 斜率0.5440, 相关性0.9992
NOTICE: 真实数据82509, 预测数据82187, 本次预测偏差%0.3900
NOTICE: 样本数量1, 截距4622.2503, 斜率0.9465, 相关性0.8993
NOTICE: 真实数据92670, 预测数据111447, 本次预测偏差%20.2600
NOTICE: 样本数量1, 截距4531.6850, 斜率0.9481, 相关性0.9003
NOTICE: 真实数据112865, 预测数据145539, 本次预测偏差%28.9500
NOTICE: 样本数量412, 截距19211.5611, 斜率0.8778, 相关性0.9590
NOTICE: 真实数据148731, 预测数据150848, 本次预测偏差%1.4200
NOTICE: 样本数量412, 截距18806.5399, 斜率0.8820, 相关性0.9580
NOTICE: 真实数据149961, 预测数据156169, 本次预测偏差%4.1400
NOTICE: 样本数量412, 截距17050.6057, 斜率0.8991, 相关性0.9603
NOTICE: 真实数据155748, 预测数据161289, 本次预测偏差%3.5600
NOTICE: 样本数量412, 截距14830.5241, 斜率0.9202, 相关性0.9607
NOTICE: 真实数据160430, 预测数据155939, 本次预测偏差%2.8000
NOTICE: 样本数量412, 截距16540.9704, 斜率0.9034, 相关性0.9574
NOTICE: 真实数据153344, 预测数据150240, 本次预测偏差%2.0200
NOTICE: 样本数量412, 截距17692.1060, 斜率0.8917, 相关性0.9532
NOTICE: 真实数据147997, 预测数据140772, 本次预测偏差%4.8800
NOTICE: 样本数量414, 截距41717.5731, 斜率0.6980, 相关性0.9736
NOTICE: 真实数据138023, 预测数据137013, 本次预测偏差%0.7300
NOTICE: 样本数量414, 截距42191.4454, 斜率0.6933, 相关性0.9722
NOTICE: 真实数据136535, 预测数据135075, 本次预测偏差%1.0700
NOTICE: 样本数量414, 截距42836.5141, 斜率0.6866, 相关性0.9716
NOTICE: 真实数据133978, 预测数据133909, 本次预测偏差%0.0500
NOTICE: 样本数量414, 截距42868.4919, 斜率0.6863, 相关性0.9698
NOTICE: 真实数据132634, 预测数据136891, 本次预测偏差%3.2100
NOTICE: 样本数量414, 截距39356.6117, 斜率0.7213, 相关性0.9849
NOTICE: 真实数据136998, 预测数据137674, 本次预测偏差%0.4900
NOTICE: 样本数量418, 截距-98886.5041, 斜率1.7965, 相关性0.9925
NOTICE: 真实数据136303, 预测数据136431, 本次预测偏差%0.0900
NOTICE: 样本数量414, 截距41274.0892, 斜率0.7011, 相关性0.9848
NOTICE: 真实数据130987, 预测数据130817, 本次预测偏差%0.1300
NOTICE: 样本数量414, 截距41537.1100, 斜率0.6983, 相关性0.9803
NOTICE: 真实数据127722, 预测数据129731, 本次预测偏差%1.5700
NOTICE: 样本数量414, 截距35567.9284, 斜率0.7625, 相关性0.9901
NOTICE: 真实数据126303, 预测数据124949, 本次预测偏差%1.0700
NOTICE: 样本数量414, 截距41599.7365, 斜率0.6944, 相关性0.9993
NOTICE: 真实数据117218, 预测数据117405, 本次预测偏差%0.1600
NOTICE: 样本数量413, 截距1686.3033, 斜率1.1262, 相关性0.8957
NOTICE: 真实数据109160, 预测数据110726, 本次预测偏差%1.4300
NOTICE: 样本数量412, 截距-126088.7154, 斜率2.7998, 相关性0.9671
NOTICE: 真实数据96823, 预测数据97097, 本次预测偏差%0.2800
NOTICE: 样本数量408, 截距36426.6219, 斜率0.5003, 相关性0.9205
NOTICE: 真实数据79716, 预测数据72776, 本次预测偏差%8.7100
NOTICE: 样本数量408, 截距29915.3284, 斜率0.5522, 相关性0.9813
NOTICE: 真实数据72658, 预测数据69530, 本次预测偏差%4.3100
NOTICE: 样本数量409, 截距30542.2158, 斜率0.5286, 相关性0.9970
NOTICE: 真实数据71739, 预测数据71377, 本次预测偏差%0.5100
NOTICE: 样本数量408, 截距21294.1724, 斜率0.6206, 相关性0.9985
NOTICE: 真实数据77243, 预测数据76786, 本次预测偏差%0.5900
NOTICE: 样本数量1, 截距4921.7169, 斜率0.9414, 相关性0.8898
NOTICE: 真实数据89412, 预测数据109386, 本次预测偏差%22.3400
NOTICE: 样本数量406, 截距-771730.9711, 斜率6.1383, 相关性0.9650
NOTICE: 真实数据110972, 预测数据112269, 本次预测偏差%1.1700
NOTICE: 样本数量388, 截距15580.3852, 斜率0.9001, 相关性0.9520
NOTICE: 真实数据144014, 预测数据149237, 本次预测偏差%3.6300
NOTICE: 样本数量388, 截距14377.9729, 斜率0.9129, 相关性0.9524
NOTICE: 真实数据148483, 预测数据151688, 本次预测偏差%2.1600
NOTICE: 样本数量388, 截距13455.2553, 斜率0.9226, 相关性0.9497
NOTICE: 真实数据150405, 预测数据156324, 本次预测偏差%3.9400
NOTICE: 样本数量402, 截距71505.3386, 斜率0.5561, 相关性0.9759
NOTICE: 真实数据154850, 预测数据155607, 本次预测偏差%0.4900
NOTICE: 样本数量388, 截距11270.4334, 斜率0.9451, 相关性0.9387
NOTICE: 真实数据151244, 预测数据144638, 本次预测偏差%4.3700
NOTICE: 样本数量388, 截距14060.3682, 斜率0.9147, 相关性0.9332
NOTICE: 真实数据141118, 预测数据129415, 本次预测偏差%8.2900
NOTICE: 样本数量390, 截距36957.1231, 斜率0.7099, 相关性0.9656
NOTICE: 真实数据126106, 预测数据125617, 本次预测偏差%0.3900
NOTICE: 样本数量390, 截距37150.1505, 斜率0.7077, 相关性0.9636
NOTICE: 真实数据124896, 预测数据128489, 本次预测偏差%2.8800
NOTICE: 样本数量390, 截距34760.7660, 斜率0.7330, 相关性0.9714
NOTICE: 真实数据129061, 预测数据128477, 本次预测偏差%0.4500
NOTICE: 样本数量390, 截距35229.1317, 斜率0.7280, 相关性0.9667
NOTICE: 真实数据127849, 预测数据128208, 本次预测偏差%0.2800
NOTICE: 样本数量392, 截距4342.9938, 斜率1.0018, 相关性0.9702
NOTICE: 真实数据127715, 预测数据129117, 本次预测偏差%1.1000
NOTICE: 样本数量393, 截距-32076.9878, 斜率1.3312, 相关性0.9964
NOTICE: 真实数据124554, 预测数据124206, 本次预测偏差%0.2800
NOTICE: 样本数量393, 截距-19541.0766, 斜率1.2152, 相关性1.0000
NOTICE: 真实数据117397, 预测数据117404, 本次预测偏差%0.0100
NOTICE: 样本数量390, 截距47549.0400, 斜率0.5872, 相关性0.9902
NOTICE: 真实数据112693, 预测数据111435, 本次预测偏差%1.1200
NOTICE: 样本数量390, 截距50098.4943, 斜率0.5560, 相关性0.9977
NOTICE: 真实数据108804, 预测数据108821, 本次预测偏差%0.0200
NOTICE: 样本数量390, 截距50042.2813, 斜率0.5567, 相关性0.9964
NOTICE: 真实数据105623, 预测数据105973, 本次预测偏差%0.3300
NOTICE: 样本数量1, 截距5273.1579, 斜率0.9358, 相关性0.8782
NOTICE: 真实数据100474, 预测数据89115, 本次预测偏差%11.3100
NOTICE: 样本数量1, 截距5280.4763, 斜率0.9354, 相关性0.8785
NOTICE: 真实数据89591, 预测数据72087, 本次预测偏差%19.5400
NOTICE: 样本数量384, 截距30325.0273, 斜率0.5354, 相关性0.9387
NOTICE: 真实数据71422, 预测数据64918, 本次预测偏差%9.1100
NOTICE: 样本数量386, 截距37631.4820, 斜率0.4029, 相关性0.9941
NOTICE: 真实数据64616, 预测数据64377, 本次预测偏差%0.3700
NOTICE: 样本数量384, 截距20707.7226, 斜率0.6191, 相关性0.9961
NOTICE: 真实数据66389, 预测数据65428, 本次预测偏差%1.4500
NOTICE: 样本数量384, 截距17341.5766, 斜率0.6472, 相关性0.9978
NOTICE: 真实数据72238, 预测数据72772, 本次预测偏差%0.7400
NOTICE: 样本数量1, 截距5202.6036, 斜率0.9363, 相关性0.8805
NOTICE: 真实数据85644, 预测数据102774, 本次预测偏差%20.0000
NOTICE: 样本数量382, 截距-211937.8855, 斜率2.3700, 相关性0.9232
NOTICE: 真实数据104207, 预测数据107341, 本次预测偏差%3.0100
NOTICE: 样本数量363, 截距10473.2297, 斜率0.9328, 相关性0.9381
NOTICE: 真实数据134716, 预测数据144319, 本次预测偏差%7.1300
NOTICE: 样本数量363, 截距8082.7467, 斜率0.9608, 相关性0.9426
NOTICE: 真实数据143484, 预测数据153571, 本次预测偏差%7.0300
NOTICE: 样本数量379, 截距90033.9242, 斜率0.4106, 相关性0.9539
NOTICE: 真实数据151426, 预测数据150648, 本次预测偏差%0.5100
NOTICE: 样本数量363, 截距3555.4288, 斜率1.0121, 相关性0.9344
NOTICE: 真实数据147628, 预测数据148068, 本次预测偏差%0.3000
NOTICE: 样本数量377, 截距-22855.0642, 斜率1.3040, 相关性0.9608
NOTICE: 真实数据142781, 预测数据143858, 本次预测偏差%0.7500
NOTICE: 样本数量363, 截距8135.3139, 斜率0.9564, 相关性0.9081
NOTICE: 真实数据127852, 预测数据116232, 本次预测偏差%9.0900
NOTICE: 样本数量363, 截距11650.2051, 斜率0.9095, 相关性0.9209
NOTICE: 真实数据113022, 预测数据107993, 本次预测偏差%4.4500
NOTICE: 样本数量366, 截距43850.9025, 斜率0.5911, 相关性0.9231
NOTICE: 真实数据105932, 预测数据109352, 本次预测偏差%3.2300
NOTICE: 样本数量366, 截距41459.4112, 斜率0.6193, 相关性0.9421
NOTICE: 真实数据110807, 预测数据111147, 本次预测偏差%0.3100
NOTICE: 样本数量366, 截距41099.7207, 斜率0.6234, 相关性0.9330
NOTICE: 真实数据112531, 预测数据107192, 本次预测偏差%4.7400
NOTICE: 样本数量366, 截距45144.8340, 斜率0.5733, 相关性0.9910
NOTICE: 真实数据106011, 预测数据105444, 本次预测偏差%0.5400
NOTICE: 样本数量366, 截距45652.0542, 斜率0.5670, 相关性0.9907
NOTICE: 真实数据105170, 预测数据104365, 本次预测偏差%0.7600
NOTICE: 样本数量368, 截距57233.9599, 斜率0.4495, 相关性0.9969
NOTICE: 真实数据103554, 预测数据103401, 本次预测偏差%0.1500
NOTICE: 样本数量368, 截距58816.4609, 斜率0.4327, 相关性0.9999
NOTICE: 真实数据102706, 预测数据102719, 本次预测偏差%0.0100
NOTICE: 样本数量366, 截距45837.1316, 斜率0.5648, 相关性0.9874
NOTICE: 真实数据101460, 预测数据101473, 本次预测偏差%0.0100
NOTICE: 样本数量366, 截距45788.3201, 斜率0.5655, 相关性0.9787
NOTICE: 真实数据98505, 预测数据97660, 本次预测偏差%0.8600
NOTICE: 样本数量1, 截距5430.0126, 斜率0.9322, 相关性0.8723
NOTICE: 真实数据91734, 预测数据83227, 本次预测偏差%9.2700
NOTICE: 样本数量1, 截距5423.3347, 斜率0.9320, 相关性0.8726
NOTICE: 真实数据83453, 预测数据66847, 本次预测偏差%19.9000
NOTICE: 样本数量360, 截距30435.2931, 斜率0.4928, 相关性0.9223
NOTICE: 真实数据65904, 预测数据59957, 本次预测偏差%9.0200
NOTICE: 样本数量360, 截距25313.6494, 斜率0.5394, 相关性0.9738
NOTICE: 真实数据59911, 预测数据58046, 本次预测偏差%3.1100
NOTICE: 样本数量360, 截距22789.5261, 斜率0.5623, 相关性0.9761
NOTICE: 真实数据60677, 预测数据57848, 本次预测偏差%4.6600
NOTICE: 样本数量360, 截距14289.6380, 斜率0.6383, 相关性1.0000
NOTICE: 真实数据62350, 预测数据62309, 本次预测偏差%0.0700
NOTICE: 样本数量1, 截距5349.6991, 斜率0.9328, 相关性0.8743
NOTICE: 真实数据75224, 预测数据94495, 本次预测偏差%25.6200
NOTICE: 样本数量1, 截距5276.8974, 斜率0.9345, 相关性0.8761
NOTICE: 真实数据95563, 预测数据124211, 本次预测偏差%29.9800
NOTICE: 样本数量339, 截距10611.8990, 斜率0.9248, 相关性0.9273
NOTICE: 真实数据127277, 预测数据132503, 本次预测偏差%4.1100
NOTICE: 样本数量339, 截距9346.7583, 斜率0.9398, 相关性0.9270
NOTICE: 真实数据131802, 预测数据141158, 本次预测偏差%7.1000
NOTICE: 样本数量354, 截距45602.8301, 斜率0.6964, 相关性0.9378
NOTICE: 真实数据140259, 预测数据142531, 本次预测偏差%1.6200
NOTICE: 样本数量354, 截距22118.2940, 斜率0.8984, 相关性0.9995
NOTICE: 真实数据139179, 预测数据139044, 本次预测偏差%0.1000
NOTICE: 样本数量339, 截距7066.6202, 斜率0.9646, 相关性0.9085
NOTICE: 真实数据130151, 预测数据123330, 本次预测偏差%5.2400
NOTICE: 样本数量352, 截距788258.5127, 斜率-6.1054, 相关性0.9259
NOTICE: 真实数据120531, 预测数据120243, 本次预测偏差%0.2400
check_predict
------------------------
(165320,170611,0.0320)
(167663,165922,0.0104)
(161071,150145,0.0678)
(145570,136026,0.0656)
(133155,133953,0.0060)
(133962,134241,0.0021)
(130484,127596,0.0221)
(126182,123864,0.0184)
(122998,124578,0.0128)
(125052,125119,0.0005)
(123000,121572,0.0116)
(119991,118710,0.0107)
(116182,118363,0.0188)
(116887,116082,0.0069)
(108562,98517,0.0925)
(99168,82725,0.1658)
(82505,74942,0.0917)
(73450,72804,0.0088)
(75681,76143,0.0061)
(82509,82187,0.0039)
(92670,111447,0.2026)
(112865,145539,0.2895)
(148731,150848,0.0142)
(149961,156169,0.0414)
(155748,161289,0.0356)
(160430,155939,0.0280)
(153344,150240,0.0202)
(147997,140772,0.0488)
(138023,137013,0.0073)
(136535,135075,0.0107)
(133978,133909,0.0005)
(132634,136891,0.0321)
(136998,137674,0.0049)
(136303,136431,0.0009)
(130987,130817,0.0013)
(127722,129731,0.0157)
(126303,124949,0.0107)
(117218,117405,0.0016)
(109160,110726,0.0143)
(96823,97097,0.0028)
(79716,72776,0.0871)
(72658,69530,0.0431)
(71739,71377,0.0051)
(77243,76786,0.0059)
(89412,109386,0.2234)
(110972,112269,0.0117)
(144014,149237,0.0363)
(148483,151688,0.0216)
(150405,156324,0.0394)
(154850,155607,0.0049)
(151244,144638,0.0437)
(141118,129415,0.0829)
(126106,125617,0.0039)
(124896,128489,0.0288)
(129061,128477,0.0045)
(127849,128208,0.0028)
(127715,129117,0.0110)
(124554,124206,0.0028)
(117397,117404,0.0001)
(112693,111435,0.0112)
(108804,108821,0.0002)
(105623,105973,0.0033)
(100474,89115,0.1131)
(89591,72087,0.1954)
(71422,64918,0.0911)
(64616,64377,0.0037)
(66389,65428,0.0145)
(72238,72772,0.0074)
(85644,102774,0.2000)
(104207,107341,0.0301)
(134716,144319,0.0713)
(143484,153571,0.0703)
(151426,150648,0.0051)
(147628,148068,0.0030)
(142781,143858,0.0075)
(127852,116232,0.0909)
(113022,107993,0.0445)
(105932,109352,0.0323)
(110807,111147,0.0031)
(112531,107192,0.0474)
(106011,105444,0.0054)
(105170,104365,0.0076)
(103554,103401,0.0015)
(102706,102719,0.0001)
(101460,101473,0.0001)
(98505,97660,0.0086)
(91734,83227,0.0927)
(83453,66847,0.1990)
(65904,59957,0.0902)
(59911,58046,0.0311)
(60677,57848,0.0466)
(62350,62309,0.0007)
(75224,94495,0.2562)
(95563,124211,0.2998)
(127277,132503,0.0411)
(131802,141158,0.0710)
(140259,142531,0.0162)
(139179,139044,0.0010)
(130151,123330,0.0524)
(120531,120243,0.0024)
(100 rows)
Prediction curve:
Prediction error:
Based on simple linear regression analysis to analyze Moutai's daily closing price, the following is the next day’s closing price prediction and checksum chart.
1) http://www.math.zju.edu.cn/ligangliu/Courses/MathematicalModeling_2005-2006/Syllabus/chapter_10.pdf
2) http://210.28.216.200/cai/tongji/html/main.htm
3) http://zh.wikipedia.org/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95
4) http://baike.baidu.com/view/145440.htm
5) http://baike.baidu.com/view/139822.htm
6) http://en.wikipedia.org/wiki/Simple_linear_regression
7) http://docs.oracle.com/cd/B28359_01/server.111/b28286/functions139.htm
8) http://www.postgresql.org/docs/9.4/static/functions-aggregate.html
9) http://v.ku6.com/playlist/index_6598382.html
10) http://cos.name/tag/%E5%9B%9E%E5%BD%92%E5%88%86%E6%9E%90/
PostgreSQL: Performing Linear Regression Analysis to Predict Future Data (Part 2)
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